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Wednesday, 16 October 2013

As
I discussed in my previous post, a rapid series of modulations brings
us to the topic of understanding when, at the end of §138,
the interlocutor says:

But we understand the meaning
of a word when we hear or say it; we grasp the meaning at a stroke,
and what we grasp in this way is surely something different from the
'use' which is extended in time!

This
is probably the strongest objection so far to Wittgenstein's account
of meaning. Its power stems from the way it ties into various
extremely tempting ideas about the process of communication. Above
all, it suggests that understanding is some kind of thing that
we acquire when we learn the meaning of a word, and which is
subsequently represented to us whenever we hear that word.

The
appeal of this idea is bolstered by various simple reflections. When
we come to know the meaning of a word we gain understanding –
so we must've gained something! Likewise, if we don't know the
meaning of a word then we lack understanding – so
“understanding” must be whatever it is we lack. And although we
might exhibit understanding in performance (eg, by using a word
correctly), that is not understanding itself; it is merely
evidence from which others can infer that we do indeed possess
understanding.

Moreover,
when we hear words we are immediately aware of our understanding. The
words strike us in a quite vivid and particular way. We don't simply
hear sounds or see ink-marks on a page (or shapes on a computer
screen) – we hear (or read) language. We experience
the meanings that it conveys. To see that this is so, just compare
the case of reciting a passage we don't understand (having learnt it
parrot-fashion, perhaps) with that of reciting one whose words are
familiar to us. Surely it is undeniable that what happens inside us
in the two cases is completely different?

Such
considerations present a challenge for Wittgenstein. When we
understand a word we understand its meaning. But what we grasp in the
instant of understanding doesn't seem to be anything like a use.
(And, in any case, is use something that could
be grasped in an instant?) But if what we grasp isn't use then
meaning can't be use.

2.
Pictures and their application

Wittgenstein
begins his assault on this appealing conception by considering the
notion that understanding is a picture
that comes before our mind when we hear a word (§139).
It's a natural enough place to start given that his early philosophy
propounded what's called “the picture theory of language”.
Moreover, that theory was itself a refinement of a venerable
philosophical position dating back at least as far as Locke's Essay
Concerning Human Understanding.
In my last
post I gave a rough sketch of the account of communication that
the picture theory tends to suggest. It might be helpful to repeat
that here:

Person
A has the thought, in the form of a picture, that things are
thus-and-so. She wants to communicate her thought to B so she
“converts” it into a proposition: she speaks, writes, uses
sign-language or Morse code, etc. The words she uses relate to the
elements of her picture and are arranged in a similar fashion. Next,
B perceives A's proposition and he re-converts it into a thought.
That is to say, the words he hears (reads, etc) produce a picture in
his mind. That picture is his understanding of what A said. If he has
the same picture he has understood correctly; if he has a different
picture then he has misunderstood (and if he has no picture at all
then he has not understood at all). The correctness of his picture
can be inferred from his behaviour. Of course, we can't be sure
B's picture matches A's but so long as he fetches the right object
(etc) then it is reasonable to assume that it does.

In
attacking the idea of understanding as a picture, it is this last
part of the account that Wittgenstein focuses upon: the link between
the picture and subsequent behaviour. He points out (§139)
that, whatever might happen in our minds, our behaviour is still a
criterion for understanding. If someone consistently uses
a word incorrectly then we say that he or she doesn't understand its
meaning. So understanding is clearly bound up with use in some sense.
But if understanding is a mental picture then what is the link
between that picture and use? “Can what we grasp at a
stroke agree with a use, fit or
fail to fit it?” (§139.)
Wittgenstein suggests an answer to his own question: let's assume
that when I hear the word “cube” I get a picture of a cube in my
mind. If I point to a cube my use fits the picture, but if I point to a
triangular prism instead then my use has failed to fit the picture.

At
first blush this seems straightforward, but Wittgenstein argues that
it will not do. The problem is: who is to say that pointing to the
triangular prism actually is
an incorrect use of the picture?

His objection is couched in terms of the picture's “method of projection”.
I assume that's a glance back to Wittgenstein's time as an
engineering student and refers to techniques used in technical
drawing. I'm afraid I have roughly zero understanding of such things
(I looked up geometric projection on Wikipedia and it didn't help)
but there are other ways of making the same point. Basically it comes
to this: a picture by itself stands in need of a method of
application. If no method is
stipulated then it's impossible to say whether a picture has been
used rightly or wrongly.

So,
returning to the example of “cube”, the word presents me with a
picture:

I
then look about me and spot a triangular prism. I realise that in
both cases they have equal ends and parallel rectilinear figures and
that their sides are parallelograms. I therefore point to the
triangular prism because, in that sense,
it is the same as my picture. In other words, I have used my picture
as an example of a prism (for a cube is a prism too). That might seem
an unusual way of
applying the picture, but who is to say that it's wrong?

To
see how far use can deviate from expectation, consider the following
example: I present someone with a dog and an iPhone. I then give him
a photo of the dog and ask him to choose which of the two items it
most resembles. He chooses the iPhone because, like the photo, you
can hold it in your hand, put it in your pocket, etc. Of course most
of us would automatically choose the dog, but that is only because we
are already familiar with the activity of picking things out from
photos. It is something we’ve done countless times and so it
probably wouldn’t even occur to us to use the photo in a different
way. But there is nothing that says choosing the dog must
be the correct response in all circumstances. Indeed to someone with
a different upbringing from ours the dog might seem an absurd
choice.

The
picture, don't forget, is not being cast as an aid
to understanding; it is supposed to be the thing itself.
But it's hard to see how it can play that role when it provides no
standard of correctness. If this observation seems familiar, that's
because it is closely analogous to the point made about ostensive
definition in relation to meaning. There, it was supposed that
the sample by itself
could establish a link between word and object, that it was
completely unambiguous and therefore unmistakable. But it turned out that it
only functioned as part of an established practice of describing the
rule for the use of a
word. And it's a very similar story in the case of the picture (which
is, after all, a kind of mental
sample). We have the sample, but what we lack is the application. (I
should also mention that as well as looking back to ostensive
definition this point also anticipates aspects of the discussion of
rule-following. See, for example, §201:
“if every course of action can be brought into accord with the
rule, then it can also be brought into conflict with it. And so there
would be neither accord nor conflict here”. This overlap should not
be surprising; the concept of understanding is internally connected
to the concept of meaning. Each helps define the other. And they are
both closely bound up with the concept of rule-following.)

3. Compulsion

The
reminder that a picture can be used in various different ways also
serves to highlight (and undermine) another temptation regarding
understanding as an “inner” phenomenon: the assumption that the
relation between representation and what is represented somehow takes
care of itself. We don't need to stipulate a method of application
because the picture does it for us. It's especially easy to think
like this when we focus on the immediacy and fluency with which we
usually understand language. We don't have to struggle to grasp the
meaning – in fact, it's impossible for us not
to understand. It's as if the picture somehow carries its meaning
within it, like a kind of spirit, and exposure to the picture
transfers this spirit to us, so that we cannot help but see the
picture as an image of this
object. The words (or rather the images they produce) force
an application on us (§140).

In
the Tractatus this
seems to be regarded as a kind of logical
compulsion. So long as picture and fact share the same logical
structure there cannot
be doubt about what's represented. Of course, this idea is exploded
by the simple observations in §139
(it is, in fact, another example of a rule
misrepresented as a necessary feature of the world). But there
are other ways of presenting the idea of compulsion. As Wittgenstein
says in §140,
“we might also be inclined to express ourselves like this: we're at
most under a psychological, not a logical, compulsion”. He then
adds a typically cryptic coda: “And now, indeed, it looks as if we
knew of two kinds of case.” What's he getting at here?

I
think this is a warning against defending a preconception by taking
refuge in theoretical explanations. We assumed we were under
compulsion; the idea that it was a logical one has proved empty, so
now we say it must be “some other kind”. And, happily, we hit
upon the notion of “psychological compulsion” as a
plausible-sounding alternative. (Already we can see the bewitching
idea of a mental mechanism
looming on the horizon.) But notice how vague all this is! How much
do we actually know about psychological compulsion? Is it clear
that this is an example of it? And how do we propose to find out?
Will we be conducting field experiments or can we decide things from
our armchairs? Isn't the idea of psychological compulsion just a
guess – and a guess
that invokes a mysterious realm of mental structures, subconscious
computation, and so forth?

There's
a second, perhaps more fundamental, issue. We've been attempting to
clarify the concept of understanding, but now it looks as if we're
sliding into a quasi-causal explanation. But that's like trying to
find out what a watch is
by examining the structure of its cogs and springs. Such an
investigation may be useful in various ways, but it won't tell you
anything about the role watches play in people's lives. For that you
need to describe their use,
not their internal mechanisms. The notion of compulsion tempts us
away from a conceptual investigation towards an ersatz scientific one
which is doomed to failure from the outset.

Well,
Wittgenstein may be alluding
to such thoughts in §140,
but he doesn't elaborate on them at this point. Instead, he sidesteps
the quagmire of compulsion and brings things down to earth: “our
'belief that the picture forced a particular application upon us'
consisted in the fact that only the one case and no other occurred to
us.” This, of course, is a move from the theoretical back to the
descriptive. Why no
other case occurred to us is not Wittgenstein's concern. But the fact
that they didn't made it seem like the picture could only be applied
in one way even though (as we now see) that's not true at all.

It
is this observation that subsequently allows him to answer the
question in §139:
how can what's grasped in an instant fit or fail to fit a use?
“[T]hey can clash in so far as the picture makes us expect a
different use; because people in general apply this
picture like this”
(§141).
What seemed to be a mysterious (perhaps impossible) relation between
two entirely different phenomena turns out to be remarkably simple
and ordinary.

4. Building in the
application

Maybe,
however, compulsion isn't the only answer. Perhaps we could build the
application into the picture itself, thereby allowing it to perform
its allotted function. Wittgenstein considers this in §141.
“How,” he asks, “am I to imagine this?” The answer he
suggests is a picture of two cubes with lines of projection between
them. Something a bit like this perhaps:

Actually,
I don't think that quite works. The cube on the left presumably
represents the mental image and the one on the right represents
actual cubes in the world (and the lines attempt to show the relation
between the two). But we no longer have the mental image that the
left-cube pictures! So we're going to need two separate images:
first, the picture of the cube by itself and, second, the picture of
the two cubes showing how the cube in the first picture is to be
applied. But this raises a further problem: how do we know the
relationship between these two images? How do we know that the
left-cube in the second image represents the cube in the first image?
Aren't we going to need a third
image depicting the relation between the two we already have? And
won't that in turn require a fourth and fifth image linking the third
image to the first and second ones? And so on.

In
a roundabout way this comes to the same objection that Wittgenstein
makes about his own proposed image: whatever its content it will
still just be another picture, and will therefore stand in need of an
application. It's not simply that the initial picture didnot provide its
application (as if that were a kind of oversight); it could
not do so. Application cannot be
provided in that way.

Is
Wittgenstein claiming, then, that there's no such thing as an
application coming before one's mind? Of course not, but (he reminds
us in §141)
understanding what that
amounts to involves looking at the use of the phrase “the
application came before his mind”. It does not
involve a doomed attempt to posit a hypothetical picture with
miraculous powers. Instead of analysing the phrase himself, however, Wittgenstein leaves it up to us investigate. I think it's worth
having a go because the conclusions it suggests (to me, at any rate)
tie in with many of the points he will shortly be making about other
accounts of understanding. Here's what I came up with:

a. Suppose I've been
teaching someone to use mathematical formulae. Now I show her the
series 0, 1, 1, 2, 3, 5, 8 and ask her to continue it. She ponders
for a while and then cries “I've got it!” I ask her what's
occurred to her and she writes down the formula “Fn = Fn-1 + Fn-2”.
Then she correctly continues the series. That would count, I think,
as an example of the application coming before her mind. Something
occurred to her and it helped her continue the series.

b. However, it would also
count if, instead of the formula, she writes down “Each new number
is the sum of the previous two”. So what comes before her mind
needn't be one specific thing.

c.
But suppose instead when I show her the series she simply says “Oh,
that's easy!” and correctly continues it: “...13, 21, 34, 55...”.
And when I ask what occurred to her she says “Nothing – it was
obvious.” (Perhaps she's worked through the series numerous times
before.) Here no
application came before her mind; she just knew what to do. In other
words, correct performance by itself is not a sufficient criterion.
Did she understand my
order to continue the series? Yes, but that shows you can understand
something without an
application coming before your mind.

d.
Finally, suppose that she writes down the formula as in the first
example, but then has no idea how to continue the series. It later
transpires that she'd previously seen the formula written below the
series and that's why it occurred to her. Here again the application
has not come before her mind even though what occurs to her – the
formula – is exactly the same as in the first example. So correct
performance might not be sufficient, but it's clearly important.

But
what, then, is the difference between (a) and (d)? From the point of
view of the criteria for the phrase “the application came before
her mind”, the difference is not what's in her mind but her past
experiences: her education, training and so on. The formula only
counts as the application in the right circumstances.
We might sum up all these reflections in the following way: however
you slice it, understanding involves more (and sometimes less) than
an application coming before one's mind.

5. Conclusion

A picture cannot fulfil
the role of understanding by itself. This is not to deny that at
least sometimes a word can bring an image before our minds, nor that
the image might help us to use the word correctly. But it cannot
show that we have used the word correctly; it supplies no
method of application and hence no standard of correctness – and
the criteria for applying the term “understanding” are closely
bound up with correct usage.

However, the appeal of
supposing understanding to be an inner “Something” is not so
easily dispelled. Thus, for example, we might think that (questions
of application aside) the pictorial account of understanding is
implausible simply because it doesn't seem to match what actually
happens. When we hear someone speak we don't get a torrent of
pictures running through our minds one after the other. Yet instead
of making us abandon the idea, this objection typically leads to
attempts at refining it. We might wonder, for example, if the
pictures are somehow unnoticed. Do they go by too fast? Are they
perhaps unconscious? And need they be conventional images at
all? Mightn't they take a form that we wouldn't easily recognise as
pictures – formulae, for example? Now comes the thought that what
we're really talking about is something that allows us to compute
the correct output for a given input. And if that's the case, then
isn't “understanding” a question of having the right mental
structure – one that facilitates correct responses? And need
this structure actually be mental? Couldn't it be a state of
the brain? And so on.

It is these types of
“refinement” that Wittgenstein considers in the remaining
sections on understanding. They all, I think, flounder on the point
already made: “understanding” is bound up with application, and
correctness of application can only be given by an established
practice. Nevertheless, it is important for Wittgenstein to
deal with these new variations because otherwise the suspicion will
remain that he has been over-hasty. The prejudice in favour of the
inner “Something” is deep-rooted; digging it out requires tracing
it through all its manifestations. So that's the next post.

Thursday, 10 October 2013

§138
begins a sustained investigation into the concept of understanding.
I'll begin unpicking its various strands (which is a daunting
prospect) in my next post, but here I want to examine the way
Wittgenstein introduces the topic. Basically §138
perplexed me when I re-read it recently and it took a fair amount of
effort to tease out the trajectory of its thought. That might just
have been dimness on my part (in which case I apologise in advance
for what's to follow), but I suspect that for many readers of the
Investigations – especially those unfamiliar with the
Tractatus – it can be easy to miss what Wittgenstein is
doing in this section.

Typically,
the shift between propositional
form (§§134-137) and understanding (§§138-184) is treated as
one of those abrupt changes of subject the author warns us about in
his preface. Fogelin, for example, states that in §138 “Wittgenstein
turns from his attack upon Tractarian themes” (Fogelin,
Wittgenstein, second edition, p144 – I like the word
“Tractarian”, by the way, and intend to make full use of it from
now on). Hacker and Baker, by contrast, are a bit more nuanced. In
Wittgenstein: Understanding and Meaning we are told that
“§§1-142 of the Investigations are Janus-faced, looking
back to the errors of the Tractatus […] and forward to the
very different account now being unfolded” (Volume I, p357). And
then, a little later, “§§138–142
mark a change” (ibid). So, it would seem, the breach-proper comes
at §143 and §§138-142 form a sort of
pivot or gateway into the full expression of Wittgenstein's new
philosophical approach. Nonetheless, even here there's little or no
sense that the discussion of understanding arises
out of what comes before it, nor are
we offered an account of how or why the one leads to the other.
(Hacker and Baker might cover this in the exegesis, but – forgive
me – I've never read it. There are
limits, you know.)

But
that's curious, because when you actually read the text the
transition feels pretty seamless. Usually when Wittgenstein makes a
jump it's not too hard to spot (eg, §134 or §243), but in this case
§138 is clearly presented as a continuation
of the discussion in §137. Indeed, the topic of understanding is
raised before we've hardly realised it. At the same time, however, it
must be admitted that there's something a bit odd about the
transition. It reads
like a continuation and yet, upon closer inspection, it's not at all
clear how it fits with what's gone before. Is Wittgenstein resorting
to a kind of authorial sleight-of-hand? Why would he bother –
especially given his near-pathological aversion to fudging things?
Why not just make the jump and have done with it? Time for a closer
look.

§§136-137
consider the claim that the concept of truth and falsity
(truth-functionality) fits
the concept “proposition”. That is to say, the two concepts
necessarily
go together. Wittgenstein exposes this idea as a grammatical remark
misleadingly disguised as a description (see Propositional
Form). At first blush, the start of §138
merely seems part of this debate; with typical doggedness the
interlocutor suggests a further example of “fitting” in order to
shore up his position:

“But can’t the
meaning of a word that I understand fit the sense of a sentence that
I understand? Or the meaning of one word fit the meaning of another?”

When
you think about it, though, this is strange. What does the idea of a
word “fitting” the sense of a sentence have to do with
truth-functionality “fitting” propositions? How does the one
support the other? It seems a glaring non sequitur. Of course,
they both use the notion of “fitting” in relation to language –
but, frankly, so what? If that’s all it amounts to then the
interlocutor might just as well have said “But surely there’s
such a thing as words fitting (or failing to fit) on a page?” We
need something stronger than that.

To
make matters worse, Wittgenstein's response is in some ways just as
puzzling: “Of course, if the meaning is the use we make of
the word, it makes no sense to speak of such fitting” (§138).
This seems to suggest that if meaning is use then there's no such
thing as a word fitting the sense of a sentence or of one word
fitting the meaning of another. But this is surely wrong! Not only do
we often talk about words fitting in this way, but when we do we are
usually concerned precisely with the question of use.

So,
for example, someone with little English who wanted a drink might
wonder whether it was correct to say “Pass me the jug” or “Pass
me the water”. In such a case we would tell them that it didn't
really matter; in this context the words “jug” and “water”
both fitted the sense of the sentence. The two expressions
achieve the same thing; they are used in the same way.
(Imagine a culture where, instead of “jug” or “water”, they
had a word meaning “water-in-jug” that was always used on such
occasions. So if you said “Pass me the jug” they'd empty out the
water before handing it over, and if you said “Pass me the water”
they'd pour the water into your hands. In such a situation neither
“jug” nor “water” would fit the sense of the sentence – and
again that is entirely due to the conventions governing use.)
Here “fitting” is akin to aptness; it's a question of
which words are effective, which words get the job done.

The
final puzzle in this puzzling section comes with the interlocutor's
response to Wittgenstein's criticism, which suddenly shifts the focus
away from “sense” and on to “understanding”. With palpable
exasperation the interlocutor says:

But we understand
the meaning of a word when we hear or say it; we grasp the meaning at
a stroke, and what we grasp in this way is surely something different
from the 'use' which is extended in time!

This
is a pivotal moment in the Investigations and I'll be
discussing it in detail in my next post. But for our present purposes
the important issue is the connection between sense and
understanding. It may or may not be true to say that the phenomenon
of grasping meaning “at a stroke” sits uneasily with the claim
that meaning is use, but how on earth is this bound up with the idea
of a word fitting the sense of a sentence?

So
now we have three questions:

How
does the idea of a word fitting the sense of a sentence support the
idea of truth-functionality fitting the concept of a proposition?

Why
does Wittgenstein respond by seemingly ruling out the idea of such
fitting on the grounds that meaning is use?

What
is the connection between the sense of a sentence and understanding?

The
answer to all three, I think, lies in what the interlocutor means by
“the sense of a sentence”. In common usage, explaining the sense
of a sentence typically involves paraphrasing it into a form that's
easier to understand. In such a context, therefore, “sense” is
roughly synonymous with “meaning”. But it shouldn't be forgotten
that §§134-137 deal explicitly with
the Tractarian notion of propositional form, so when the interlocutor
raises the subject of “sense” in §138 he is still looking at
things from the point of view of the Tractatus.
That is, he is using “sense” according to the definition given at
TLP 2.221: “What a picture represents is its sense”.

This
brings us up against the picture
theory of language. According to the Tractatus, the
essential thing about language is that it pictures logically
possible states of affairs. And it does this by mirroring the logical
structure of what it represents. So “The cat is on the mat”
presents us with objects arranged in a particular way, just like a
drawing might. The assertion it makes may or may not be true – to
find out we’d have to compare the sentence-picture with reality.
But even if it’s false it still has a sense because what it
pictures is possible: cats can be on mats. Compare this, however, to
“The cat is on selfishness”. That proposition is nonsense
precisely because it attempts to combine phenomena in an illicit way;
selfishness can’t be sat on any more than it can be eaten or set on
fire. So what the proposition attempts to represent cannot even be
pictured and therefore lacks sense.

Now,
how can we tell which words can legitimately be combined with which?
That is down to the a priori structure of the world, for it is this
structure which determines the combinational possibilities of
objects. And these combinational possibilities must, in turn, be
reflected in the logical syntax governing language. It is as if a
sentence beginning “The cat is on…” leaves a gap which can only
be filled by words of the right type. Or, to put it another way, only
the right type of word will fit. This, I think, is the type of
“fitting” that the interlocutor has in mind at the start of §138.
It belongs to a conception of language as a structure made up
of different shaped holes into which only the right kind of
“word-peg” can be slotted. And this type of “fitting” has
nothing to do with aptness or use; it is modelled on the physical
sense of the word, as when a plug fits into a socket or a hand fits
into a glove. Its ultimate legitimacy comes from the world rather
than linguistic convention.

Now
consider the very next line of the Tractatus, TLP 2.222:

The agreement or
disagreement of its sense with reality constitutes its truth or
falsity.

A
proposition with a sense must be a picture of a possible state
of affairs. As such, that possible state of affairs must
either obtain or fail to obtain. In other words, a proposition must
be either true or false. So, according to the Tractatus, the
very notion of sense necessarily brings with it the notion of truth
and falsity. You cannot have one without the other.

Now
I think we can begin to see the connection between §137 and §138.
So far as the interlocutor is concerned, the requirement for
propositions to be true or false arises out of the definition of
“sense”. Criticism of the former inevitably implies criticism of
the latter. It is as if the interlocutor – worried by the attack on
truth-functionality – moves back a step to protect the prior link
in the chain of reasoning. For if that holds good then (it seems) the
subsequent links must also be preserved.

To
put it more generally, both claims are part of the same broad
conception of language, ie: language as a kind of calculus whose
rules reflect the necessary structure of existence and rigidly
determine the way words operate. The various elements of the
conception are closely interwoven (that is part of the beauty of the
Tractatus); they hold each other in place and, consequently,
if any one element is attacked the others automatically rise up in
its defence (this, I think, sheds light on Wittgenstein's remark in
the Blue Book (p44) that “no philosophical problem can be
solved until all philosophical problems are solved”).

The
answer to question (2) is also now clear. It is not “fitting” per
se that Wittgenstein objects to so much as the interlocutor's
specific interpretation of the term and the philosophical theory that
lies behind it. (We might say that the interlocutor's remark at the
beginning of §138 is somewhat
duplicitous – he is trading on the ordinary meaning of “fitting”
and “sense” to make his claim seem straightforward and
uncontroversial.) The Tractarian account of “fitting” does indeed
make no sense if meaning is use. For as we have seen, from the
viewpoint of use, “fitting” is a matter of the role the word
plays rather than its ability to mirror a supposed a priori
world-structure. There is such a thing as a word fitting the
sense of a sentence, only not in the way the interlocutor claims.

Are
we any closer to answering question (3)? I think we are, but only if
we read between the lines. The interlocutor's exclamation at the end
of §138 (quoted above) marks
the point where we move from discussing ideas explicitly set out in
the Tractatus and on to what we might call the “uncredited
assumptions” that lurk behind those ideas and help make them seem
plausible. And one of those assumptions concerns the link between
sense and understanding.

To
clarify this we need to return to the picture theory of language. As
we've seen, according to that theory a proposition pictures a
possible state of affairs which is its sense. But, of course, a
proposition doesn't just come out of nowhere – it expresses a
thought, and a thought is itself a picture (TLP 3: “A
logical picture of facts is a thought”). Therefore “In a
proposition a thought finds an expression that can be perceived by
the senses” (TLP 3.1). Now, what is it to understand a
proposition? 4.024 tells us: “To understand a proposition means to
know what is the case if it is true”.

From
these brief remarks it is possible to construct an account of
communication that goes like this: A has the thought that things are
thus-and-so. And this thought is in some sense a picture – that is,
whatever form it takes it must combine objects in a way that mirrors
a possible state of affairs. A wants to communicate her thought to B
so she “converts” it into a proposition. She speaks, writes,
draws, uses sign-language or Morse code, etc. The elements of the
proposition are arranged in a way that mirrors their combination in
A's thought (and thus also mirror a possible state of affairs in the
world). Now B perceives A's proposition and he (so to speak)
re-converts it into a thought in his mind. And that thought is, of
course, a picture. The sense of the picture is what it
represents, and it is precisely this that B needs to grasp in order
to understand the proposition. Grasping the sense of the picture
(seeing that it represents this possible state of affairs) and
understanding the proposition are one and the same thing.

This
account will be familiar to anyone who's studied Empirical philosophy
(especially Locke). The connection it makes between sense and
understanding couldn't be more direct: grasping the sense is
understanding the proposition. Understanding, therefore, is a thing
(specifically, a picture) that we acquire instantaneously when
we perceive the proposition. And that is surely right, isn't it? For
when we hear or read words we understand them at once; we don't have
to wait until we've studied their use before we know what they
mean. So if the sense of a sentence isn't to be cashed-out along
Tractarian lines how do we account for instantaneous understanding?

That
thought, it seems to me, completes the chain of argument that leads
from the discussion of propositional form in §137
to the topic of understanding at the end of §138.
Wittgenstein's criticism of “fitting” in relation to
truth-functionality causes the interlocutor to defend the notion of
sense from whence the necessity of truth-functionality originated.
And criticism of the notion of sense in turn provokes an appeal to
the nature of understanding, because the immediacy of understanding
can only be understood (it seems) if something like the Tractarian
account of sense holds good. By now we have indeed moved away from
the explicit theories of the Tractatus (which is notoriously
silent about how thought, understanding etc actually work), and so in
a way Fogelin, Hacker et al are right: §138
represents a break. But it shouldn't be thought that we've moved on
to completely unrelated issues. We have moved from the explicit
theories of the Tractatus to the implicit assumptions that lay
behind them.